In this section, we remind the definition of the polytope ring associated with a convex
polytope \(P\subset \mathbb {R}^n\) .
This ring was originally introduced by Khovanskii and Pukhlikov in [1]} to give
a convenient functorial description of the cohomology (or Chow) rings of smooth toric
varieties.
Later, Kaveh used Khovanskii–Pukhlikov ring to give a partial description of cohomology
rings of spherical varieties [2]}.
Recently, Khovanskii–Pukhlikov rings were applied to cohomology rings of toric bundles
[3]}
and Schubert calculus on polyhedral realizations of Demazure crystals [4]}.
